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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be a field, let $n \\ge 1$, and let $A$ be an $n \\times n$ [[matrix|0JXZKX526R]] over $\\mathbb{K}$.\nA *square root* of $A$ is an $n \\times n$ matrix $B$ over $\\mathbb{K}$ such that $B^2 = A$, where $B^2 = BB$ is the [[matrix product|032Q9FB8MT]].\n\n## Remarks\n\nExample: for every real $\\theta$, $B = \\begin{pmatrix} \\cos\\theta \u0026 \\sin\\theta \\\\ \\sin\\theta \u0026 -\\cos\\theta \\end{pmatrix}$ is a square root of $I_2$.\nCheck: the entries of $B^2$ are $\\cos^2\\theta + \\sin^2\\theta = 1$ on the diagonal and $\\cos\\theta\\sin\\theta - \\sin\\theta\\cos\\theta = 0$ off it.\nNon-example: $N = \\begin{pmatrix} 0 \u0026 1 \\\\ 0 \u0026 0 \\end{pmatrix}$ has no square root.\nIf $B^2 = N$, then $B^4 = 0$, so $B$ is [[nilpotent|07BSPZPSMF]]; a nilpotent $2 \\times 2$ matrix satisfies $B^2 = 0 \\ne N$.\nA [[positive semidefinite matrix|01MMVW8CKC]] has a unique positive semidefinite square root, written $A^{1/2}$; see [[square root of a positive semidefinite matrix|0BJVDCF38Y]].\nA complex matrix with no eigenvalue in $(-\\infty, 0]$ has a unique square root whose eigenvalues have positive real part, the *principal square root*.\n",
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